A first order phase transition in the threshold-$\theta \ge 2$ contact process on random $r$-regular graphs and $r$-trees

Abstract

We consider the discrete-time threshold-$\theta \ge 2$ contact process on a random r-regular graph on n vertices. In this process, a vertex with at least θoccupied neighbors at time t will be occupied at time t+1 with probability p, and vacant otherwise. We show that if $\theta \ge 2$ and $r\ge \theta +2$, ${\epsilon }_1$ is small and p is at least $p_1({\epsilon }_1)$, then starting from all vertices occupied the fraction of occupied vertices stays above $1-2{\epsilon }_1$ up to time $\exp ({\gamma }_1(r)n)$ with probability at least $1-\exp (-{\gamma }_1(r)n)$. In the other direction, we show that for $p_2<1$ there is an ${\epsilon }_2(p_2)>0$ so that if $p\le p_2$ and the number of occupied vertices in the initial configuration is at most ${\epsilon }_2(p_2)n$, then with high probability all vertices are vacant at time $C_2(p_2)\log (n)$. These two conclusions imply that on the random r-regular graph there cannot be a quasi-stationary distribution with density of occupied vertices between 0 and ${\epsilon }_2(p_1)$, and allow us to conclude that the process on the r-tree has a first order phase transition.

20 pages